Highly efficient slab-loaded microwave resonator design: analytical solution and numerical validation

Abstract

The critical element of a microwave heating system is the resonator, which affects the effective heating of the material to be heated. It is based on the creation of an effective electric field on the material. In order to create a homogeneous and strong electric field in the entire volume of the material to be heated, it is necessary to design a resonator of the correct dimensions. In this study, a model is derived in which the required resonator dimensions for a microwave heating system with a specific resonance frequency can be calculated analytically. The design of a rectangular microwave resonator that resonates at 2.45 GHz is given purely analytically using a derived model. The extracted analytical model is validated by simulation results for two different modes.

Appendix

Phase constant of free space:

$${\text{k}}_{{\text{o}}} {\text{ = (w}}\sqrt {{(}\mu_{0} \varepsilon_{0} {)}} {)}$$

(12)

Cutoff frequencies of the transverse electric (TE) and transverse magnetic (TM) modes:

$$\begin{array}{*{20}c} {f_{{c_{1} }} = (c/2)\sqrt {((m/A)^{2} + (n/B)^{2} )} } & {m,n = 0,1,2,...} \\ \end{array}$$

(13)

The phase constants for the resonator and the waveguide are as shown in the following equations.

$$\beta = (k_{o} \sqrt {(1 - (f_{{c_{1} }} /f)^{2} )} )$$

(14)

$${\text{tan}}\delta {\text{ = (image(}}\varepsilon_{r} {\text{)/real(}}\varepsilon_{r} {))}$$

(15)

The loss tangent can be expressed as above.

$$\gamma = (k_{o} \sqrt {((f_{c} /f)^{2} - real(\varepsilon_{r} )(1 - i(\tan \delta )))} ) = \alpha ^{\prime} + i\beta ^{\prime}$$

(16)

$\gamma$ is the propagation constant of the loaded resonator; ($\alpha ^{\prime}$ is the attenuation constant, and $\beta ^{\prime}$ is the phase constant)

$$\begin{gathered} \alpha ^{\prime} = real(\gamma ) \hfill \\ \beta ^{\prime} = imag(\gamma ) \hfill \\ \end{gathered}$$

(17)

For TE type modes, the wave impedance for the material and the free space of the resonator can be defined as follows:

$$Z_{er} = \left( {{1 \mathord{\left/ {\vphantom {1 {\sqrt {\left( { - \left( {{{f_{c1} } \mathord{\left/ {\vphantom {{f_{c1} } f}} \right. \kern-0pt} f}} \right)^{2} + real\left( {\varepsilon_{r} } \right)\left( {1 - i\left( {\tan \;\delta } \right)} \right)} \right)} }}} \right. \kern-0pt} {\sqrt {\left( { - \left( {{{f_{c1} } \mathord{\left/ {\vphantom {{f_{c1} } f}} \right. \kern-0pt} f}} \right)^{2} + real\left( {\varepsilon_{r} } \right)\left( {1 - i\left( {\tan \;\delta } \right)} \right)} \right)} }}} \right)$$

(18)

$$z_{1} = (1/\sqrt {(1 - (f_{c1} /f)^{2} )} )$$

(19)

where z_er is the wave impedance of the material, and z_l is the wave impedance of the empty space of the resonator.